Prof. Dr.

Ali Hameed Aziz Theory of Structures

2- Stability and Determinacy of Structures

2-1- Stability and Determinacy of Beams
The Beams is a horizontal member which transfer loads to supports (columns or walls). Figure (10) shows
the common types of beams.

Propped cantilever

Figure (10) Common types of beams

The total Equations of Equilibrium for any beam are three:-

Σ𝐹𝑥=0
Σ𝐹𝑦=0
Σ𝑀=0
In addition to the above equations, there are Equations of Conditions (C). Therefore, the total equations
of equilibrium = C+3

To check the determinacy and stability of beams, the general law is: r 3+C

Where
r =Total number of reaction in beam (unknowns).
C= Equations of conditions (if any).
If:
1- r ˂ 3+C → The beam is unstable.
2- r = 3+C→ The beam is stable and determinate.
3- r ˃ 3+C→ The beam is stable and indeterminate to (D=r-(3+C)) Degree.

Examples

Beam r C

4 1 4=1+3, Stable Determinate

5 2 5=2+3, stable determinate

4 2 4<2+3, Unstable Internally

3 - Unstable externally (Reactions are concurrent)

7 2 7>2+3, Stable indeterminate, 2nd degree

6 2 6>2+3, Stable indeterminate, 1st degree

7 2 7>2+3, Unstable internally

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 Prof. Dr. Ali Hameed Aziz Theory of Structures

3 2 3<2+3, Unstable

Homework No. (1)

Discuss the stability and determinacy of the following structures.

2-2- Stability and Determinacy of Frames

The frames are structures often used in buildings, industrial establishments, warehouses,….etc.; and are
composed of beams and columns that are either pin or fixed connected. Generally, the loading on frames
causes three types of forces on its members; Axial Force (N) works parallel to the member axis, Shear
Force (V) works perpendicular to the member axis, and Bending Moment (M).

Figure (11) Axial Force (N), Shear Force (V) and Bending Moment (M).

There are two types of frames, Opened Frames (opened bay frame) and Closed Frames (closed bays
frame).
2-2-1- Stability and Determinacy of Opened Frames
In this case, the frames is treated same as a beams. To check the determinacy and stability of the
opened frame the general law is: r 3+C

r=9 E=3
r=5 E=3 r˃E
r˃E r=9, C=1, E=3
Stable and r˃E
Stable and +Indeterminate Stable and
+Indeterminate to 6th Degree +Indeterminate
to 2nd Degree to 5th Degree

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 Prof. Dr. Ali Hameed Aziz Theory of Structures

r=3 E=3 C=1 r=4 E=3 C=2

r=2 E=3
r ˂ E+C r ˂ E+C
r˂E
Internally unstable Internally unstable
Externally unstable (Mechanism)
(Reactions are parallel) (Mechanism)

Figure (11) Common types of opened frames

2-2-2- Stability and Determinacy of Closed Frames

To check the determinacy and stability of the closed frame the general law is:

3b+ r 3j+C

Where:
b = No. of members
r = No. of reactions
j = No. of joint
C = Equations of conditions (if any).

If:
1- 3b+r ˂ 3j+C → The frame is unstable.
2- 3b+r = 3j+C→ The frame is stable determinate.
3- 3b+r ˃ 3j+C→ The frame is stable indeterminate to (D=3b+r-(3j+C)) Degree.

Examples

Frame b r j c

39>27
10 9 9 0
Stable Indeterminate, 12th Degree

51>43
14 9 13 4
Stable Indeterminate, 8th Degree

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 Prof. Dr. Ali Hameed Aziz Theory of Structures

39>30
10 9 9 3* Stable Indeterminate, 9th Degree
*See page (6)

42>31
11 9 10 1
Stable Indeterminate, 11th Degree

The cantilever is not 38>34

considered a member

11 5 10 4* Stable Indeterminate, 4th Degree

*See page (6)

Homework No. (2)

Discuss the stability and determinacy of the following structures.

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 Prof. Dr. Ali Hameed Aziz Theory of Structures

2-3- Stability and Determinacy of Plane Trusses

A truss may be defined as a structure which consists of a number of straight members pin-connected
together at their ends so as to form a rigid structure. In practice, the truss members are bolted, riveted or
welded at their ends; but, for structural analysis, the truss member assumed to be pin-connected. The
trusses are often used in bridges, buildings, industrial establishments, warehouses, hangars,….etc.

Assumption for ideal Truss

1-All joints are frictionless pins.
2-External load and reactions are only applied at the pin joints.
3-All the truss members are straight and will be link member subjected to either axial tension (Ties) or
axial compression (Struts).

In trusses, at each joint there are two equations of equilibrium:-

Σ𝐹𝑥=0
Σ𝐹𝑦=0

To check the determinacy and stability of truss, the general law is: r+b 2J

Where
r =Total number of reactions.
b=Total number of bars (members).
J= Total number of joints.

If
1- b+r < 2j → The truss is unstable
2- b+r = 2j→ The truss is stable determinate
3- b+r > 2j → The truss is stable indeterminate
Important Note: It may be noted that, the above mentioned equations is not always sufficient to decide
whether the truss is stable or not!!

Truss b r j

10=10
7 3 5
Stable Determinate

10=10
7 3 5
Unstable Internally

10=10
6 4 5
Unstable Internally

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 Prof. Dr. Ali Hameed Aziz Theory of Structures

16=16
13 3 8
Unstable Internally

12=12
9 3 6
Unstable Internally

17>16
Links 14 3 8
Unstable Externally (Parallel Reactions)

18>16
15 3 8
Stable Indeterminate, 2nd Degree

19˃18
15 4 9
Stable Indeterminate, 1st Degree

Homework No. (3)

Discuss the stability and determinacy of the following structures.

2-4- Stability and Determinacy of Arches

An arch is a curved beam or structure subjected to loads act on the convex side of the curve and re-sights
the external loads by the force of thrust. It is subjected to three restraining forces, Axial Thrust Force (N)
acting with the arch axis; Shear Force (V) and Bending Moment (M). The arches are often used in
bridges, industrial establishments, hangars,….etc. The arches can be classified based on their boundary
conditions (supports).

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 Prof. Dr. Ali Hameed Aziz Theory of Structures

To check the determinacy and stability of the arches, the general law is: r 3+C

Where
r =Total number of reaction in beam (unknowns).
C= Equations of conditions (if any).

r=2 E=3 r=3 E=3

r˂E r =E
Externally Unstable Stable +Determinate
(Reaction are parallel)

r=3 E=3 C=1 r=4 E=3 C=2

r ˂ E+C r ˂ E+C
Internally Unstable Internally Unstable
(Mechanism) (Mechanism)

The common types of the arches are:-

1- Fixed at both ends with no hinges present at its crown.
2- Fixed at both ends with a hinge at its crown.
3- Two-hinged arches.
4- Three-hinged arches

Figure (12) Common types of arches

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